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http://dx.doi.org/10.5666/KMJ.2020.60.2.255

Divide Knot Presentation of Knots of Berge's Sporadic Lens Space Surgery  

Yamada, Yuichi (Dept. of Mathematics, The University of Electro-Communications)
Publication Information
Kyungpook Mathematical Journal / v.60, no.2, 2020 , pp. 255-277 More about this Journal
Abstract
Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a sequel of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot. In the present paper, L-shaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples of Berge's lens space surgery, is presented by a generalized L-shaped curve as a divide knot. A formula on the surgery coefficients and the presentation is also considered.
Keywords
Dehn surgery; lens space; plane curve;
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1 Y. Yamada, Canonical forms of the knots in the genus one fiber surfaces, Bull. Univ. Electro-Commun., 22-1(2010), 25-31.
2 V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps, Volume II. Monodromy and asymptotics of integrals, Monographs in Mathematics 83, Birkhauser Boston, Inc., Boston, MA, 1988.
3 K. L. Baker, Knots on Once-punctured torus fibers, Ph. D. dissertation, The University of Texas at Austin, 2004.
4 K. L. Baker, Surgery descriptions and volumes of Berge knots I: Large volume Berge knots, J. Knot Theory Ramifications, 17(9)(2008), 1077-1197.   DOI
5 K. L. Baker, Surgery descriptions and volumes of Berge knots II: Descriptions on the minimally twisted five chain link, J. Knot Theory Ramifications, 17(9)(2008), 1099-1120.   DOI
6 J. Berge, Some knots with surgeries yielding lens spaces, Unpublished manuscript, 1990.
7 J. Berge, The knots in $D^2{\times}S^1$ which have nontrivial Dehn surgeries that yield $D^2{\times}S^1$, Topology Appl., 38(1)(1991), 1-19.   DOI
8 S. Chmutov, Diagrams of divide links, Proc. Amer. Math. Soc., 131(5)(2003), 1623-1627.   DOI
9 O. Couture and B. Perron, Representative braids for links associated to plane immersed curves, J. Knot Theory Ramifications, 9(2000), 1-30.   DOI
10 A. Deruelle, K. Miyazaki and K. Motegi, Networking Seifert surgeries on knots. II. The Berge's lens surgeries, Topology Appl., 156(6)(2009), 1083-1113.   DOI
11 A. Deruelle, K. Miyazaki and K. Motegi, Networking Seifert surgeries on knots, Mem. Amer. Math. Soc., 217(2012), no. 1021, 130 pp.
12 R. Fintushel and R. Stern, Constructing Lens spaces by surgery on knots, Math. Z., 175(1980), 33-51.   DOI
13 C. McA Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Math. Soc. Japan, (1991), 631-642.
14 C. V. Q. Hongler and C. Weber, The link of an extrovert divide, Ann. Fac. Sci. Toulouse Math. (6), 9(1)(2000), 133-145   DOI
15 C. McA Gordon, Dehn filling: a survey, Knot theory, Banach Center Publ., 42(1998), 129-144.   DOI
16 J. E. Greene, The lens space realization problem, Ann. of Math. (2), 177(2)(2013), 449-511.   DOI
17 M. Hirasawa, Visualization of A'Campo's fibered links and unknotting operation, Topology Appl., 121(2002), 287-304.   DOI
18 R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of 3-manifolds III, Trans. Amer. Math., 234(1977), 245-251.
19 L. Rudolph, Knot theory of complex plane curves, Handbook of Knot Theory, W W. Menasco and M B. Thistlethwaite Eds, Elsevier B. V., Amsterdam, (2005), 349-427.
20 Y. Yamada, Berge's knots in the fiber surfaces of genus one, lens space and framed links, J. Knot Theory Ramifications, 14(2)(2005), 177-188.   DOI
21 H. Goda, M. Hirasawa and Y. Yamada, Lissajous curves as A'Campo divides, torus knots and their fiber surfaces, Tokyo J. Math., 25(2)(2002), 485-491.   DOI
22 Y. Yamada, A family of knots yielding graph manifolds by Dehn surgery, Michigan Math. J., 53(3)(2005), 683-690.   DOI
23 Y. Yamada, Finite Dehn surgery along A'Campo's divide knots, Singularity Theory and its Applications, Adv. Stud. Pure Math., 43(2006), 573-583.   DOI
24 Y. Yamada, Lens space surgeries as A'Campo's divide knots, Algebr. Geom. Topol., 9(1)(2009), 397-428.   DOI
25 N. A'Campo, Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. (6), 8(1999), 5-23.   DOI
26 N. A'Campo, Le groupe de monodromie du deploiement des singularite isolees de coubes planes I, Math. Ann., 213(1975), 1-32.   DOI
27 N. A'Campo, Generic immersions of curves, knots, monodromy and gordian number, Inst. Hautes Etudes Sci. Publ. Math., 88(1998), 151-169.   DOI
28 N. A'Campo, Planar trees, slalom curves and hyperbolic knots, Inst. Hautes Etudes Sci. Publ. Math., 88(1998), 171-180.   DOI